Between songs at a recent gig, the three-piece Daughter were jovially heckled: "Where are you from?" The young band rehearse in London's King's Cross, their soon-to-be-released first album was recorded just north in Archway and chunks of it were written in Dorset. The singer, Elena Tonra, is from London's suburbs and the guitarist, Igor Haefeli, is Swiss. Drummer Remi Aguilella is from a town in central France and came to the capital by way of Fargo in uppermost America – so the question needed a long answer. Handy, because Haefeli had a tricky bit of retuning to do.

We meet in a pub in east London, not far from where that recent gig took place. My assumption as to where these 23-year-olds met (a global convention for people with cool names) is wrong. They came together at music college in 2010, in a songwriting class. Tonra had constructed a spooky, guitar-led number called Tomorrow but felt it lacked something, "some atmospheric shit". Happily, this turned out to be a speciality of her classmate, Haefeli, a music tech enthusiast, and they started to work together.

Later on, keen to gig, they recruited two more college peers, Aguilella on drums and a bass player. Tonra can't read or write music so she tried to fudge it, describing what she wanted from the band in her song with written-out instructions. Not the preferred method for the bassist – "The fear in his eyes!" – who never performed with them again. Aguilella, smartly, binned the instructions and improvised. A three-piece was forged.

They called themselves Daughter and, in 2011, released a pair of EPs. Last spring, they were signed by indie label 4AD and that composition from the songwriting class, Tomorrow, will appear as the central track on their debut album, If You Leave, out next month.

Tonra, who writes their lyrics, is in a relationship with Haefeli. "I admired his weird brain," she says, while Aguilella politely sits out this part of the conversation. Have there been awkward moments? Songs performed on stage that were about the bloke to her direct right? "I don't hold anything back with my writing and I don't think Igor wants me to. Things can sometimes overlap, but I see [the music and the relationship] quite separately, so I write what I want. It doesn't affect anything."

Any more said on this point, I suspect, might risk the childish energy that the band strive for. Their two EPs to date were called The Wild Youth and His Young Heart and both had pictures of Tonra as a kid on the cover. Her lyrics tend to deal with first lessons, with unworldliness, often sung in a vulnerable near-whisper. I was particularly taken with last year's skin-prickling track, Medicine, which explored naive decision-making.

The band's name has a certain resonance, too, but Tonra says it was adopted because she had done some "awful solo stuff" before aligning with Haefeli and Aguilella and she wanted to mark a distinction. "I didn't want it to be my name – me and two guys in the background. This is a collaboration."

Last year, Daughter were asked to appear on Late Show With David Letterman (3.5 million nightly viewers) ahead of a US tour. It was a hardly plausible offer for such an untried band – the fact that Daughter share management links with Mumford & Sons might have helped – but the trio weren't keen. "We didn't even have an album written yet, so it felt like burning a lot of stages," says Haefeli, who speaks excellent English, albeit with an endearing habit of muddling our cliches.

Eventually they were persuaded. "Making their network television debut... Daughter!" said Letterman last October, handing over to Tonra in heavy fringe and buttoned-up blouse, Aguilella with his shirt sleeves rolled up and Haefeli in an ill-fitting hat. The trio looked very young, which probably suited them. Anyway, they were excellent. "We are the reckless," purred Tonra in the eerie, half-sung chorus, "we are the wild youth."

Fittingly, the band credit any early feats – a decent gig at the prestigious Other Voices festival last month, for instance – to an attitude that has more than a little playground logic to it. "You can try hard to be heard and it has the opposite effect," Haefeli says. "If you're immersed in your own sound and don't care what other people think, quite often they'll end up listening."